Abstract
A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections ∇ and ∇ * on M. When M is dually flat, that is flat with respect to ∇ and ∇ * , a canonical divergence is known, which is uniquely determined from ( M , g , ∇ , ∇ * ) . We propose a natural definition of a canonical divergence for a general, not necessarily flat, M by using the geodesic integration of the inverse exponential map. The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. Finally, we show that the integrability of the inverse exponential map implies the geodesic projection property.
Highlights
IntroductionA divergence function D(p q) is a differentiable real-valued function of two points p and q in a manifold M
Divergence and Dual GeometryA divergence function D(p q) is a differentiable real-valued function of two points p and q in a manifold M
We already used the inverse exponential map in our previous work [6], where we studied a different divergence function
Summary
A divergence function D(p q) is a differentiable real-valued function of two points p and q in a manifold M With equality if and only if p = q It is a distance-like function, but does not necessarily share all properties of a distance. The coefficients gDij in Equation (2) define a Riemannian metric gD. The duality of the connections holds with respect to the Riemannian metric gD in terms of the following condition:. It is an important problem to define a canonical divergence in the general case. The divergence introduced in the present article recovers the original geometry directly in terms of Equations (4)–(6), it coincides with the original canonical divergence in the dually flat case
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